List of integrals of rational functions

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Template:Trigonometry The following is a list of integrals (antiderivative functions) of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antiderivative functions, see lists of integrals. See also trigonometric integral.

Generally, if the function sin(x) is any trigonometric function, and cos(x) is its derivative,

acosnxdx=ansinnx+C

In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.

Integrals involving only sine

sinaxdx=1acosax+C


sin2axdx=x214asin2ax+C=x212asinaxcosax+C


sin3axdx=cos3ax12a3cosax4a+C
xsin2axdx=x24x4asin2ax18a2cos2ax+C


x2sin2axdx=x36(x24a18a3)sin2axx4a2cos2ax+C


sinb1xsinb2xdx=sin((b2b1)x)2(b2b1)sin((b1+b2)x)2(b1+b2)+C(for |b1||b2|)


sinnaxdx=sinn1axcosaxna+n1nsinn2axdx(for n>2)


dxsinax=1aln|tanax2|+C


dxsinnax=cosaxa(1n)sinn1ax+n2n1dxsinn2ax(for n>1)


xsinaxdx=sinaxa2xcosaxa+C


xnsinaxdx=xnacosax+naxn1cosaxdx=k=02kn(1)k+1xn2ka1+2kn!(n2k)!cosax+k=02k+1n(1)kxn12ka2+2kn!(n2k1)!sinax(for n>0)



sinaxxdx=n=0(1)n(ax)2n+1(2n+1)(2n+1)!+C


sinaxxndx=sinax(n1)xn1+an1cosaxxn1dx


dx1±sinax=1atan(ax2π4)+C


xdx1+sinax=xatan(ax2π4)+2a2ln|cos(ax2π4)|+C


xdx1sinax=xacot(π4ax2)+2a2ln|sin(π4ax2)|+C


sinaxdx1±sinax=±x+1atan(π4ax2)+C

Integrands involving only cosine

cosaxdx=1asinax+C
cos2axdx=x2+14asin2ax+C=x2+12asinaxcosax+C
cosnaxdx=cosn1axsinaxna+n1ncosn2axdx(for n>0)
xcosaxdx=cosaxa2+xsinaxa+C
x2cos2axdx=x36+(x24a18a3)sin2ax+x4a2cos2ax+C
xncosaxdx=xnsinaxanaxn1sinaxdx=k=02k+1n(1)kxn2k1a2+2kn!(n2k1)!cosax+k=02kn(1)kxn2ka1+2kn!(n2k)!sinax
cosaxxdx=ln|ax|+k=1(1)k(ax)2k2k(2k)!+C
cosaxxndx=cosax(n1)xn1an1sinaxxn1dx(for n1)
dxcosax=1aln|tan(ax2+π4)|+C
dxcosnax=sinaxa(n1)cosn1ax+n2n1dxcosn2ax(for n>1)
dx1+cosax=1atanax2+C
dx1cosax=1acotax2+C
xdx1+cosax=xatanax2+2a2ln|cosax2|+C
xdx1cosax=xacotax2+2a2ln|sinax2|+C
cosaxdx1+cosax=x1atanax2+C
cosaxdx1cosax=x1acotax2+C
cosa1xcosa2xdx=sin(a2a1)x2(a2a1)+sin(a2+a1)x2(a2+a1)+C(for |a1||a2|)

Integrands involving only tangent

tanaxdx=1aln|cosax|+C=1aln|secax|+C
tan2xdx=tanxx+C
tannaxdx=1a(n1)tann1axtann2axdx(for n1)
dxqtanax+p=1p2+q2(px+qaln|qsinax+pcosax|)+C(for p2+q20)
dxtanax+1=x2+12aln|sinax+cosax|+C
dxtanax1=x2+12aln|sinaxcosax|+C
tanaxdxtanax+1=x212aln|sinax+cosax|+C
tanaxdxtanax1=x2+12aln|sinaxcosax|+C

Integrands involving only secant

See Integral of the secant function.
secaxdx=1aln|secax+tanax|+C
sec2xdx=tanx+C
sec3xdx=12secxtanx+12ln|secx+tanx|+C.


secnaxdx=secn2axtanaxa(n1)+n2n1secn2axdx (for n1)
dxsecx+1=xtanx2+C


Integrands involving only cosecant

cscaxdx=1aln|cscax+cotax|+C
csc2xdx=cotx+C
cscnaxdx=cscn1axcosaxa(n1)+n2n1cscn2axdx (for n1)
dxcscx+1=x2sinx2cosx2+sinx2+C
dxcscx1=2sinx2cosx2sinx2x+C

Integrands involving only cotangent

cotaxdx=1aln|sinax|+C
cotnaxdx=1a(n1)cotn1axcotn2axdx(for n1)
dx1+cotax=tanaxdxtanax+1
dx1cotax=tanaxdxtanax1

Integrands involving both sine and cosine

dxcosax±sinax=1a2ln|tan(ax2±π8)|+C
dx(cosax±sinax)2=12atan(axπ4)+C
dx(cosx+sinx)n=1n1(sinxcosx(cosx+sinx)n12(n2)dx(cosx+sinx)n2)
cosaxdxcosax+sinax=x2+12aln|sinax+cosax|+C
cosaxdxcosaxsinax=x212aln|sinaxcosax|+C
sinaxdxcosax+sinax=x212aln|sinax+cosax|+C
sinaxdxcosaxsinax=x212aln|sinaxcosax|+C
cosaxdxsinax(1+cosax)=14atan2ax2+12aln|tanax2|+C
cosaxdxsinax(1cosax)=14acot2ax212aln|tanax2|+C
sinaxdxcosax(1+sinax)=14acot2(ax2+π4)+12aln|tan(ax2+π4)|+C
sinaxdxcosax(1sinax)=14atan2(ax2+π4)12aln|tan(ax2+π4)|+C
sinaxcosaxdx=12acos2ax+C
sina1xcosa2xdx=cos((a1a2)x)2(a1a2)cos((a1+a2)x)2(a1+a2)+C(for |a1||a2|)
sinnaxcosaxdx=1a(n+1)sinn+1ax+C(for n1)
sinaxcosnaxdx=1a(n+1)cosn+1ax+C(for n1)
sinnaxcosmaxdx=sinn1axcosm+1axa(n+m)+n1n+msinn2axcosmaxdx(for m,n>0)
also: sinnaxcosmaxdx=sinn+1axcosm1axa(n+m)+m1n+msinnaxcosm2axdx(for m,n>0)
dxsinaxcosax=1aln|tanax|+C
dxsinaxcosnax=1a(n1)cosn1ax+dxsinaxcosn2ax(for n1)
dxsinnaxcosax=1a(n1)sinn1ax+dxsinn2axcosax(for n1)
sinaxdxcosnax=1a(n1)cosn1ax+C(for n1)
sin2axdxcosax=1asinax+1aln|tan(π4+ax2)|+C
sin2axdxcosnax=sinaxa(n1)cosn1ax1n1dxcosn2ax(for n1)
sinnaxdxcosax=sinn1axa(n1)+sinn2axdxcosax(for n1)
sinnaxdxcosmax=sinn+1axa(m1)cosm1axnm+2m1sinnaxdxcosm2ax(for m1)
also: sinnaxdxcosmax=sinn1axa(nm)cosm1ax+n1nmsinn2axdxcosmax(for mn)
also: sinnaxdxcosmax=sinn1axa(m1)cosm1axn1m1sinn2axdxcosm2ax(for m1)
cosaxdxsinnax=1a(n1)sinn1ax+C(for n1)
cos2axdxsinax=1a(cosax+ln|tanax2|)+C
cos2axdxsinnax=1n1(cosaxasinn1ax)+dxsinn2ax)(for n1)
cosnaxdxsinmax=cosn+1axa(m1)sinm1axnm+2m1cosnaxdxsinm2ax(for m1)
also: cosnaxdxsinmax=cosn1axa(nm)sinm1ax+n1nmcosn2axdxsinmax(for mn)
also: cosnaxdxsinmax=cosn1axa(m1)sinm1axn1m1cosn2axdxsinm2ax(for m1)

Integrands involving both sine and tangent

sinaxtanaxdx=1a(ln|secax+tanax|sinax)+C
tannaxdxsin2ax=1a(n1)tann1(ax)+C(for n1)

Integrands involving both cosine and tangent

tannaxdxcos2ax=1a(n+1)tann+1ax+C(for n1)

Integrals containing both sine and cotangent

cotnaxdxsin2ax=1a(n+1)cotn+1ax+C(for n1)

Integrands involving both cosine and cotangent

cotnaxdxcos2ax=1a(1n)tan1nax+C(for n1)

Integrands involving both secant and tangent

secxtanxdx=secx+C

Integrals with symmetric limits

ccsinxdx=0
cccosxdx=20ccosxdx=2c0cosxdx=2sinc
cctanxdx=0
a2a2x2cos2nπxadx=a3(n2π26)24n2π2(for n=1,3,5...)
a2a2x2sin2nπxadx=a3(n2π26(1)n)24n2π2=a324(16(1)nn2π2)(for n=1,2,3,...)

Integral over a full circle

02πsin2m+1xcos2n+1xdx=0{n,m}

References

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